The first sentence is kind of a strange question, by definition it's your asset as long as you own it. When you sell an asset at a profit there is a capital gain, that may be taxed. The date on which you land in the US and become an LPR is the date from which you will be liable to pay tax on income or gains. As a LPR you are taxed on worldwide income so it doesn't matter where the property is or whether or not you bring the proceeds into the US, it must be declared on your tax return. Whether or not you will actually owe any tax on it is more complicated, depending what the asset is, how much the gain is, whether you are liable for any tax in the home country and whether or not there is a double tax treaty in place.
By the way if you have more than $10k total overseas in bank accounts or stocks at any point during a year while you are a green card holder, you need to file a report (FBAR) annually listing these assets. It's not very complicated,but must be reported.